Strong non-vanishing of cohomologies and strong non-freeness of adjoint line bundles on $n$-Raynaud surfaces

TL;DR

This paper introduces n-Raynaud surfaces, explores their pathological behaviors, and uses these findings to disprove Fujita's conjecture for surfaces in positive characteristic.

Contribution

It defines n-Raynaud surfaces and demonstrates their pathological properties, providing a counterexample to Fujita's conjecture in positive characteristic.

Findings

n-Raynaud surfaces exhibit pathological behaviors

Disproof of Fujita's conjecture for surfaces in positive characteristic

Introduction of n-Tango curves and n-Raynaud surfaces

Abstract

We begin by formally defining n-Tango curves and n-Raynaud surfaces. Our investigation then focuses on the pathological behaviors exhibited by n-Raynaud surfaces. As a direct corollary of this analysis, we present a concise disproof of Fujita's conjecture for surfaces in positive characteristics.